How your P&L's average and spread grow with trade count

Question

Each independent trade has mean profit m and variance v (standard deviation \sqrt{v}). After N trades, how do the mean, variance, standard deviation, and signal-to-noise ratio of total P&L scale with N? Give the numbers for m = \2, \sqrt{v} = \50, at N = 100 and N = 10{,}000.

Accepted Answer

For a sum of N independent trades: - Mean adds: \mathbb{E}[	ext{total}] = N m, grows linearly in N. - Variance adds: 	ext{Var}[	ext{total}] = N v, grows linearly in N. - Standard deviation is the square root: \sqrt{N v} = \sqrt{N}\,\sqrt{v}, grows like \sqrt{N}. - Signal-to-noise (mean over standard deviation): \dfrac{Nm}{\sqrt{N}\sqrt{v}} = \sqrt{N}\,\dfrac{m}{\sqrt{v}}, grows like \sqrt{N}. The whole story is that the mean outpaces the standard deviation by a factor of \sqrt{N}. The dollar risk (standard deviation) does keep growing, but the edge grows faster, so relative to your profit …

How your P&L's average and spread grow with trade count

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